Asymptotic Statistics of Nodal Domains of Quantum Chaotic Billiards in the Semiclassical Limit

Quantum chaos concerns eigenfunctions of the Laplace operator in a domain where a billiard ball would bounce chaotically. Such chaotic eigenfunctions have been conjectured to share statistical properties of their nodal domains with a simple percolation model, from which many interesting quantities can be computed analytically. We numerically test conjectures on the number and size of nodal domains of quantum chaotic eigenfunctions at very high energies, approaching the semiclassical limit. We use a highly efficient scaling method to quickly compute eigenfunctions at low resolution and interpolate to higher resolution. We computed 10^5 eigenfunctions and counted 10^9 nodal domains. Our results agree with the conjectured size nodal domains but disagree with the conjectured mean and variance of the number of nodal domains